Theorem frege55lem2b 39029

Description: Lemma for frege55b 39030. Core proof of Proposition 55 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
frege55lem2b	⊢ (𝑥 = 𝑦 → [𝑦 / 𝑧]𝑧 = 𝑥)

Proof of Theorem frege55lem2b

Step	Hyp	Ref	Expression
1		frege54cor1b 39027	. 2 ⊢ [𝑥 / 𝑧]𝑧 = 𝑥
2		frege53b 39023	. 2 ⊢ ([𝑥 / 𝑧]𝑧 = 𝑥 → (𝑥 = 𝑦 → [𝑦 / 𝑧]𝑧 = 𝑥))
3	1, 2	ax-mp 5	1 ⊢ (𝑥 = 𝑦 → [𝑦 / 𝑧]𝑧 = 𝑥)

Syntax hints:   → wi 4  [wsb 2067

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-12 2220  ax-13 2389  ax-ext 2803  ax-frege8 38942  ax-frege52c 39021

This theorem depends on definitions:  df-bi 199  df-an 387  df-ex 1879  df-sb 2068  df-clab 2812  df-cleq 2818  df-clel 2821  df-sbc 3663

This theorem is referenced by:  frege55b  39030