Theorem gencl 1365

Description: Implicit substitution for class with embedded variable.

Hypotheses

Ref	Expression
gencl.1	⊢ (θ ↔ ∃x(χ ∧ A = B))
gencl.2	⊢ (A = B → (φ ↔ ψ))
gencl.3	⊢ (χ → φ)

Assertion

Ref	Expression
gencl	⊢ (θ → ψ)

Distinct variable group(s):   ψ,x

Proof of Theorem gencl

Step	Hyp	Ref	Expression
1		gencl.1	. 2 ⊢ (θ ↔ ∃x(χ ∧ A = B))
2		gencl.2	. . . . . 6 ⊢ (A = B → (φ ↔ ψ))
3		gencl.3	. . . . . 6 ⊢ (χ → φ)
4	2, 3	syl5bi 183	. . . . 5 ⊢ (A = B → (χ → ψ))
5	4	com12 13	. . . 4 ⊢ (χ → (A = B → ψ))
6	5	imp 277	. . 3 ⊢ ((χ ∧ A = B) → ψ)
7	6	19.23aiv 952	. 2 ⊢ (∃x(χ ∧ A = B) → ψ)
8	1, 7	sylbi 174	1 ⊢ (θ → ψ)

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∃wex 678   = wceq 1091

This theorem is referenced by:  2gencl 1366  3gencl 1367  indpi 3828  axrnegex 4080  axrrecex 4081

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-6 675  ax-gen 677  ax-17 925

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

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