Theorem sbimi 854

Description: Infer substitution into antecedent and consequent of an implication.

Hypothesis

Ref	Expression
sbimi.1	⊢ (φ → ψ)

Assertion

Ref	Expression
sbimi	⊢ ([y / x]φ → [y / x]ψ)

Proof of Theorem sbimi

Step	Hyp	Ref	Expression
1		sbimi.1	. . . 4 ⊢ (φ → ψ)
2	1	syl3 18	. . 3 ⊢ ((x = y → φ) → (x = y → ψ))
3	1	anim2i 270	. . . 4 ⊢ ((x = y ∧ φ) → (x = y ∧ ψ))
4	3	19.22i 723	. . 3 ⊢ (∃x(x = y ∧ φ) → ∃x(x = y ∧ ψ))
5	2, 4	anim12i 268	. 2 ⊢ (((x = y → φ) ∧ ∃x(x = y ∧ φ)) → ((x = y → ψ) ∧ ∃x(x = y ∧ ψ)))
6		df-sb 853	. 2 ⊢ ([y / x]φ ↔ ((x = y → φ) ∧ ∃x(x = y ∧ φ)))
7		df-sb 853	. 2 ⊢ ([y / x]ψ ↔ ((x = y → ψ) ∧ ∃x(x = y ∧ ψ)))
8	5, 6, 7	3imtr4 192	1 ⊢ ([y / x]φ → [y / x]ψ)

Syntax hints:   → wi 2   ∧ wa 196  ∃wex 678   = weq 797  [wsb 852

This theorem is referenced by:  bisb 855  sbi2 885  sbco 910  sbal1 996  sbal 997  tfinds2 2405

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-gen 677

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853

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