Theorem a4a 842

Description: Specialization with implicit substitution. Compare Lemma 14 of [Tarski] p. 70.

Hypotheses

Ref	Expression
a4a.1	⊢ (ψ → ∀xψ)
a4a.2	⊢ (x = y → (φ → ψ))

Assertion

Ref	Expression
a4a	⊢ (∀xφ → ψ)

Proof of Theorem a4a

Step	Hyp	Ref	Expression
1		a4a.2	. . . . 5 ⊢ (x = y → (φ → ψ))
2	1	com12 13	. . . 4 ⊢ (φ → (x = y → ψ))
3		a4a.1	. . . 4 ⊢ (ψ → ∀xψ)
4	2, 3	syl6 23	. . 3 ⊢ (φ → (x = y → ∀xψ))
5	4	19.20i 691	. 2 ⊢ (∀xφ → ∀x(x = y → ∀xψ))
6		ax9 807	. 2 ⊢ (∀x(x = y → ∀xψ) → ψ)
7	5, 6	syl 12	1 ⊢ (∀xφ → ψ)

Syntax hints:   → wi 2  ∀wal 672   = weq 797

This theorem is referenced by:  a4c 843  chv2 850  a4b 927

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-9 799

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

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