Theorem frege55lem2c 44490

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

Assertion

Ref	Expression
frege55lem2c	⊢ (𝑥 = 𝐴 → [𝐴 / 𝑧]𝑧 = 𝑥)

Distinct variable group:   𝑥,𝑧

Allowed substitution hints:   𝐴(𝑥,𝑧)

Proof of Theorem frege55lem2c

Step	Hyp	Ref	Expression
1		vex 3458	. . 3 ⊢ 𝑥 ∈ V
2	1	frege54cor1c 44488	. 2 ⊢ [𝑥 / 𝑧]𝑧 = 𝑥
3		frege53c 44487	. 2 ⊢ ([𝑥 / 𝑧]𝑧 = 𝑥 → (𝑥 = 𝐴 → [𝐴 / 𝑧]𝑧 = 𝑥))
4	2, 3	ax-mp 5	1 ⊢ (𝑥 = 𝐴 → [𝐴 / 𝑧]𝑧 = 𝑥)

Syntax hints:   → wi 4   = wceq 1560  Vcvv 3454  [wsbc 3744

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734  ax-frege8 44382  ax-frege52c 44461

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1563  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3456  df-sbc 3745  df-sn 4583

This theorem is referenced by: (None)